Boson particles quantization

The fact that every state may be occupied by several particles shows that the second quantization particles are bosons. However, in terms of different commutation relations an equivalent scheme may be obtained for fermions. To achieve this objective the wave functions are written in decomposed form as before ... 

S. Depaguit, and J. P. Vigier, Phenomenological spectroscopy of baryons and bosons considered as discrete quantized states of an internal structure of elementary particles, C. R. Acad. Sci., Ser B (Sciences Physiques) 268(9), 657-659 (1969). 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

More fundamentally, in the ballistic phonon regime at low enough temperatures, one-dimensional (ID) wires should manifest the quantization of the thermal conductance for the lowest energy modes [3,4]. Here K = JIAT is the thermal conductance with AT as the temperature difference. The fundamental quantum of thermal conductance is V klTI3h where is the Boltzmann constant, h is Planck s constant, and T is the temperature. This value is universal, independent not only of the conducting material, but also of the particle statistics, i.e., the quantum conductance is the same for bosons and fermions [3]. 

J, quantizes its component along the z axis, and II = 1 represents the parity with respect to the inversion. As to the invariance with respect to permutations of identical particles an acceptable wave function has to be antisymmetric with respect to the exchange of identical fermions, whereas it has to be s)mmetric when exchanging bosons. 

We may anticipate that the Klein-Gordon equation is not able to describe an electron for reasons given in the next section. However, in a properly quantized field-theoretical form (compare chapter 7) it describes neutral mesons of spin 0. Mesons are strongly interacting bosons, i.e., they are particles subject to the strong force (also called hadrons) with an integer spin subject to Bose-Einstein statistics (an example for a fermionic hadron would be the proton having spin 1/2). 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

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